Tricritical directed percolation with long-range interaction in one and two dimensions

Abstract: Recently, the quantum contact process, in which branching and coagulation processes occur both coherently and incoherently, was theoretically and experimentally investigated in driven open quantum spin systems. In the semi-classical approach, the quantum coherence effect was regarded as a process in which two consecutive atoms are involved in the excitation of a neighboring atom from the inactive (ground) state to the active state (excited $s$ state). In this case, both second-order and first-order transitions occur. Therefore, a tricritical point exists at which the transition belongs to the tricritical directed percolation (TDP) class. On the other hand, when an atom is excited to the $d$ state, long-range interaction is induced. Here, to account for this long-range interaction, we extend the TDP model to one with long-range interaction in the form of $\sim 1/r^{d+\sigma}$ (denoted as LTDP), where $r$ is the separation, $d$ is the spatial dimension, and $\sigma$ is a control parameter. In particular, we investigate the properties of the LTDP class below the upper critical dimension $d_c=$ min$(3,\,1.5\sigma)$. We numerically obtain a set of critical exponents in the LTDP class and determine the interval of $\sigma$ for the LTDP class. Finally, we construct a diagram of universality classes in the space ($d$, $\sigma$).